Logarithm Properties Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

medium
Use the change of base formula to evaluate log⁡320\log_3 20 to three decimal places.

Solution

  1. 1
    Apply the change of base formula: log⁡320=ln⁡20ln⁡3\log_3 20 = \frac{\ln 20}{\ln 3}.
  2. 2
    Compute: ln⁡20≈2.9957\ln 20 \approx 2.9957 and ln⁡3≈1.0986\ln 3 \approx 1.0986.
  3. 3
    log⁡320≈2.99571.0986≈2.727\log_3 20 \approx \frac{2.9957}{1.0986} \approx 2.727.

Answer

≈2.727\approx 2.727
The change of base formula log⁡ba=ln⁡aln⁡b\log_b a = \frac{\ln a}{\ln b} lets you compute logarithms in any base using a calculator's natural log or common log button.

About Logarithm Properties

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x.

Learn more about Logarithm Properties →

More Logarithm Properties Examples

Example 1 easy

Expand [formula] using logarithm properties.

Example 2 medium

Condense [formula] into a single logarithm.

Example 4 easy

Expand [formula].

← All Logarithm Properties Examples Logarithm Properties Concept